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## Stabilizer Codes And Quantum Error Correction.

## 5 Qubit Code

## To further lower the logical error rate, we turn to a family of codes known as concatenated codes.

## Contents |

Let C1 be **an [n, k1, d1] code and** let C2 be an [n, k2, d2] code. Note that for P, Q ∈ Pn, wt(PQ) ≤ wt(P) + wt(Q). Stabilizer codes have a special relationship to a finite subgroup Cn of the unitary group U(2n) frequently called the Clifford group. The smallest distance-3 CSS code is the 7-qubit code, a [[7, 1, 3]] QECC created from the classical Hamming code (consisting of all sums of classical strings 1111000, 1100110, 1010101, and 1111111). news

And for reasonably sized quantum computers, that fraction can be arbitrarily large — although the larger it is, the more qubits the computer requires. “There were many, many different proposals, all About the MIT News Office MIT News Press Center Press Inquiries Filming Guidelines Office of Communications Contact Us Terms of Use RSS Twitter Facebook Google+ Instagram Flickr YouTube MIT Homepage MIT Grassl and M. It is assumed that measurements and classical computations can be performed quickly and reliably, and that quantum gates can be performed between arbitrary pairs of qubits in the computer, irrespective of other

H. General codes[edit] In general, a quantum code for a quantum channel E {\displaystyle {\mathcal {E}}} is a subspace C ⊆ H {\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}}} , where H {\displaystyle {\mathcal This theorem seems to present an obstacle to formulating a theory of quantum error correction. The fault-tolerant procedures concatenate as well, and after L levels of concatenation, the effective logical error rate is pt(p/pt)2L (for a base code correcting 1 error).

Calderbank and P. P∣ψ⟩ = ∣ψ⟩∀P ∈ S. doi:10.1103/PhysRevA.55.1613. ^ Peter Shor (2002). "The quantum channel capacity and coherent information" (pdf). Quantum Error Correction Book Thus, the distance of the quantum code is at least min(d1, d2), but might be higher because of the possibility of degeneracy.

The position is for research in quantum error prevention methods. 5 Qubit Code There exists a stabilizer quantum error-correcting **code that achieves the** hashing limit R = 1 − H ( p ) {\displaystyle R=1-H\left(\mathbf {p} \right)} for a Pauli channel of the following The final state represents the result of the computation. pt is the threshold for fault-tolerant quantum computation.

This is the reason for quantum computers’ potential advantages: A string of qubits in superposition could, in some sense, perform a huge number of computations in parallel. Steane Code Math. the Shor code, encodes 1 logical qubit in 9 physical qubits and can correct for one bit flip and one phase flip error. The sign flip code[edit] Quantum circuit of the phase flip code Flipped bits are the only kind of error in classical computer, but there is another possibility of an error with

- In that case, let us consider tensor products of the Pauli matrices $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}, X=\begin{pmatrix}0&1\\1&0\end{pmatrix}, Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ Define the Pauli group Pn as the group consisting of tensor products of I, X,
- Available at http://arxiv.org/abs/quant-ph/0501099 G.
- We prove the theorem for this special case by exploiting random stabilizer codes and correcting only the likely errors that the channel produces.

What is more, the outcome of this operation (the syndrome) tells us not only which physical qubit was affected, but also, in which of several possible ways it was affected. Your cache administrator is webmaster. Stabilizer Codes And Quantum Error Correction. Proof. Quantum Code 7 Universal fault-tolerance is known to be possible for any stabilizer code, but in most cases the more complicated type of construction is needed for all but a few gates.

Of these, only the assumption of independent errors is at all necessary, and that can be considerably relaxed to allow short-range correlations and certain kinds of non-Markovian environments. navigate to this website A finite-depth decoding circuit corresponding to the stabilizer G {\displaystyle {\mathcal {G}}} exists by the algorithm given in (Grassl and Roetteler 2006). On a stabilizer code, therefore, logical Pauli operations can be performed via a transversal Pauli operation on the physical qubits. Britton, W. 5 Qubit Quantum Error Correction

S {\displaystyle {\mathcal {S}}} does not contain the operator − I ⊗ n {\displaystyle -I^{\otimes n}} . Lett., vol. 77, no. 5, pp. 793–797, Jul 1996. Because of the simple structure of the Pauli group, any Abelian subgroup has order 2n − k for some k and can easily be specified by giving a set of n − k commuting generators. More about the author Quantum error correction also employs syndrome measurements.

Instead of the unencoded ∣ + ⟩ state, we must use a more complex ancilla state ∣00…0⟩ + ∣11…1⟩ known as a 'cat' state. Bit Flip Memory Error R. Grassl and M.

Comput. The construction takes two binary classical linear codes and produces a quantum code, and can therefore take advantage of much existing knowledge from classical coding theory. R is called the recovery or decoding operation and serves to actually perform the correction of the state. Quantum Code Reel The basic stabilizer and its first shift are as follows: ⋯ | I I I I I I I I I I I I | X X X Z Z Z

J. However for larger N an exponentially growing number of states are possible. A transversal operation is one in which the ith qubit in each block of a QECC interacts only with the ith qubit of other blocks of the code or of special http://caribtechsxm.com/quantum-error/quantum-error-correction-usc.php A sequence has finite support if its weight is finite.

Experimental realization[edit] There have been several experimental realizations of CSS-based codes. If we only wish to detect errors, a distance d code can detect errors on up to d − 1 qubits. Brun. Therefore, if p is below the threshold pt, we can achieve an arbitrarily good error rate ε per logical gate or timestep using only poly(logε) resources, which is excellent theoretical scaling.

However, the effects of relaxing these assumptions on the threshold value and overhead requirements have not been well-studied. J. Each row of the parity check matrix can be converted into a Pauli operator by replacing each 0 with an I operator and each 1 with a Z operator. A non-degenerate code is one for which different elements of the set of correctable errors produce linearly independent results when applied to elements of the code.

Schoelkopf, "Realization of Three-Qubit Quantum Error Correction with Superconducting Circuits," Nature 482, 382-385 (2012), doi:10.1038/nature10786, arXiv:1109.4948 ^ M. Through the transmission in a channel the relative sign between | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } can become inverted. Laflamme, W. The first inequality follows, since we correct only the typical errors because the atypical error set has negligible probability mass.

Theorem 2 Let S be a stabilizer with n − k generators, and let S ⊥ = {E ∈ Pn s.t. [E, M] = 0 ∀M ∈ S}. A generalisation of this concept are the CSS codes, named for their inventors: A. Rev.